[Paper Review] Dirichlet-to-Neumann maps on bounded Lipschitz domains

Jussi Behrndt, Tom Ter Elst|arXiv (Cornell University)|Mar 13, 2014

Numerical methods in inverse problems22 references3 citations

TL;DR

This paper presents a systematic study of Dirichlet-to-Neumann and Neumann-to-Dirichlet maps on bounded Lipschitz domains using the framework of linear relations in Hilbert spaces. It establishes a Kre’in-type resolvent formula for the difference of Dirichlet and Neumann resolvents that holds for all complex parameters, even when the Dirichlet problem is not uniquely solvable, by treating the maps as multivalued relations and leveraging extension theory of symmetric operators.

ABSTRACT

The Dirichlet-to-Neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying Dirichlet problem is not uniquely solvable. The main objective of this paper is to present a systematic study of the Dirichlet-to-Neumann map and its inverse, the Neumann-to-Dirichlet map, in the framework of linear relations in Hilbert spaces. Our treatment is inspired by abstract methods from extension theory of symmetric operators, utilizes the general theory of linear relations and makes use of some deep results on the regularity of the solutions of boundary value problems on bounded Lipschitz domains.

Motivation & Objective

• To provide a systematic and intrinsic analysis of Dirichlet-to-Neumann and Neumann-to-Dirichlet maps on bounded Lipschitz domains, especially in cases where the Dirichlet problem lacks unique solvability.
• To treat the maps as linear relations in Hilbert spaces, allowing for a rigorous framework that includes multivalued behavior when eigenvalues are present.
• To establish a general Kre’in-type formula for the resolvent difference of Dirichlet and Neumann realizations of the Schrödinger operator, valid for all complex parameters.
• To utilize abstract extension theory of symmetric operators and deep regularity results on Lipschitz domains to derive spectral and mapping properties of the relations.

Proposed method

• The authors model the Dirichlet-to-Neumann map as a linear relation in $H^{1/2}({\mathcal{C}}) \times H^{-1/2}({\mathcal{C}})$, capturing both single-valued and multivalued behavior depending on the spectral parameter.
• They employ the theory of symmetric and selfadjoint linear relations in Hilbert spaces, including the use of $Q$-functions and Weyl functions, to analyze the maps.
• The paper uses trace theory on Lipschitz domains and regularity results for solutions of elliptic boundary value problems to ensure the well-definedness of the maps.
• A key technical tool is the use of the inverse of a selfadjoint relation $A^{-1}$, particularly in the context of the relation $T = B^* A^{-1} B$, which is shown to be essentially selfadjoint under suitable conditions.
• The authors derive a general resolvent difference formula between the Dirichlet and Neumann realizations via a Kre’in-type identity, valid for all $\lambda, \mu \in \mathbb{C}$, even when $\lambda$ or $\mu$ are eigenvalues.
• They establish connections between the maps $\mathcal{D}(\lambda)$ and $\mathcal{D}(\mu)$ for arbitrary $\lambda, \mu$, generalizing known results beyond the regularity regime.

Experimental results

Research questions

• RQ1 How can the Dirichlet-to-Neumann map be rigorously defined and analyzed when the Dirichlet problem is not uniquely solvable?
• RQ2 What is the structure of the Dirichlet-to-Neumann map as a linear relation in Hilbert spaces, especially when the underlying problem has nontrivial kernel?
• RQ3 Can a Kre’in-type formula for the resolvent difference of Dirichlet and Neumann realizations be extended to all complex parameters, including eigenvalues?
• RQ4 How do the spectral and mapping properties of the Dirichlet-to-Neumann and Neumann-to-Dirichlet maps depend on the geometry of the domain and the potential $V$?

Key findings

• The Dirichlet-to-Neumann map $\mathcal{D}(\lambda)$ is a selfadjoint linear relation in $H^{1/2}({\mathcal{C}}) \times H^{-1/2}({\mathcal{C}})$ for all $\lambda \in \mathbb{C}$, even when $\lambda$ is an eigenvalue of the Dirichlet realization $A_D$.
• The Neumann-to-Dirichlet map $\mathcal{N}(\lambda)$ is a selfadjoint linear relation in $H^{-1/2}({\mathcal{C}}) \times H^{1/2}({\mathcal{C}})$, and its domain and multivalued part are characterized via trace theory and regularity results.
• A general Kre’in-type formula for the resolvent difference $ (A_D - \lambda)^{-1} - (A_N - \lambda)^{-1} $ is established that holds for all $\lambda \in \mathbb{C}$, not just regular values, by interpreting the maps as linear relations.
• The maps $\mathcal{D}(\lambda)$ and $\mathcal{D}(\mu)$ are related via a formula in Theorem 4.6 that holds for all $\lambda, \mu \in \mathbb{C}$, extending known results to the multivalued regime.
• The paper proves that the relation $T = B^* A^{-1} B$ is essentially selfadjoint in $\mathcal{G}$ if $A$ is selfadjoint and $\operatorname{ran} A$ is closed, and selfadjoint if $B^*|_{\ker A}$ is boundedly invertible.